# The harmony of being at the minimum

I am always amazed by the idea of being at the minimum. Because every physical system near the minimum is an harmonic oscillator, and we love harmonic oscillator !

Let us consider a generic potential energy function $U(x)$ and suppose that $x_0$ is a stationary point for $U(x)$. A stationary point could be a minimum, maximum or inflection point of a differentiable function, in which the first derivative is null. In other words in a stationary point the function stops to increase or decrease.

If we expand in Taylor series the potential $U(x)$ in $x_0$ to the second order we have:

$U(x) = U(x_0) + \frac{1}{2}\frac{d}{dx}U(x=x_0) (x-x_0) + \frac{d^2}{dx^2}U(x=x_0) (x-x_0)^2 + o(x-x_0)^3$

Since $x_0$ is a stationary point for $U(x)$ the first derivative is null $\frac{d}{dx}U(x=x_0) = 0$ and the potential becomes:

$U(x) = U(x_0) + \frac{1}{2}\frac{d^2}{dx^2}U(x=x_0) (x-x_0)^2 + o(x-x_0)^3$

The first term $U(x_0)$ is a number equal to the value of the potential in the stationaty point. Since a generic potential energy is defined up to an additive constant, we can put $U(x_0)=0$. Furthermore, the expression $o(x-x_0)^3$ means if we stop the expansion of the potential to the second order we commit an error that is an infinitesima quantity lower than $(x-x_0)^3$ .

Finally, we can express the potential energy near a stationary point in this form:

$U(x) = \frac{1}{2}k (x-x_0)^2$

(where $k=\frac{d^2}{dx^2}U(x=x_0)$) that is the potential energy of a harmonic oscillator !

The harmonic oscillator is wherever you look in theoretical Physics (Quantum Mechanics, Quantum Field Theory , Solid State Physics and so on …).
The most important thing is we are able to exactly solve the harmonic oscillator, and then you can solve every physical system near a stationary point.

Lennard – Jones Energy Potential (source: Wikipedia)

For example, consider the Lennard – Jones potential, phenomenologically describing the short range interaction between neutral particles:

$V_{LJ}(r)=\epsilon[(\frac{r_m}{r})^{12}-2\frac{r_m}{r})^6]$

where $\epsilon$ is the depth of the potential,  $r$ is the particles relative distance and $r_m$ is the stationary point. If we expand the LJ potential near the minimum we have:

$V_{LJ}(r)=-\frac{1}{2}k(r-r_m)^2$

(where  $k=\frac{-72\epsilon}{r_0^4}$ )

When the inter-particle distance is approximately equal to $r_m$ they vibrate in harmony !

So, if you stay in the minimum of your potential energy, sin prisa, you can vibrate sin pausa (cit. R.B)